Convergence of iterated Aluthge transform sequence for diagonalizable matrices

Abstract

Given an $r\times r$ complex matrix $T$, if $T=U|T|$ is the polar decomposition of $T$, then, the Aluthge transform is defined by $$ Δ(T)= |T|^{1/2} U |T |^{1/2}. $$ Let $Δ^{n}(T)$ denote the n-times iterated Aluthge transform of $T$, i.e. $Δ^{0}(T)=T$ and $Δ^{n}(T)=Δ(Δ^{n-1}(T))$, $n\in\mathbb{N}$. We prove that the sequence $\{Δ^{n}(T)\}_{n\in\mathbb{N}}$ converges for every $r\times r$ {\bf diagonalizable} matrix $T$. We show that the limit $Δ^{\infty}(\cdot)$ is a map of class $C^\infty$ on the similarity orbit of a diagonalizable matrix, and %of class $C^\infty$ on the (open and dense) set of $r\times r$ matrices with $r$ different eigenvalues.